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\title[ch05]{Chapter 06: Differential Equations}
\author[]{SCC LQW}
%\institute[XX大学]{XX大学\quad 数学与统计学院\quad 数学与应用数学专业}
%\date{2025年6月}

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% 封面页
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  \titlepage
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% 目录页
\begin{frame}{Contents}
  \tableofcontents
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 1
\section{THE D-MODULE OF AN EQUATION.}
%---------------------------------------------------
\begin{frame}{1.1 DEFINITION. }

Let $P_1, \ldots, P_m$ be differential operators in $A_n$. 

Then we have a system of differential equations

\begin{equation}
    P_1(f) = \cdots = P_m(f) = 0.
    \tag{1.1}
\end{equation}

The {\color{red}$A_n$-module associated to the system of differential equations} (1.1) is $$A_n / \sum_{i=1}^{m} A_n P_i. $$


\noindent\rule{\textwidth}{0.4pt}


\end{frame}


%---------------------------------------------------
\begin{frame}{1.2 THEOREM.}

Let $M$ be the $A_n$-module associated with the system (1.1). 

The vector space of {\color{red}polynomial solutions} of the system (1.1) is isomorphic to $$\text{Hom}_{A_n}(M, K[X]).$$

Let $\mathcal{S}$ be a left $A_n$-module; and let $M$ be a finitely generated left $A_n$-module. 

We will call $$\text{Hom}_{A_n}(M, \mathcal{S})$$ the {\color{red}solution space} of $M$ in $\mathcal{S}$. 

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{1.3 EXAMPLE.}

Note that we have taken care \underline{not} to require that $\mathcal{S}$ be finitely generated. 

For example, $C{\,}^{\infty}(\mathbb{R}^n)$ is not a finitely generated $A_n(\mathbb{R})$-module. 

The advantage of a definition of this sort is that it allows us to introduce generalized solutions of differential equations in a natural way. 

All one has to do is choose an appropriate $A_n$-module $\mathcal{S}$. 

This includes solutions in terms of distributions, hyperfunctions and microfunctions. 

\noindent\rule{\textwidth}{0.4pt}


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 2
\section{DIRECT LIMIT OF MODULES.}
%---------------------------------------------------
\begin{frame}{2.1 DEFINITION. }

We say that $\{M_i : i \in \mathcal{I}\}$ is a {\color{red}directed family} of left $R$-modules if given any $i,j \in \mathcal{I}$, satisfying $i \leq j$, there exists a homomorphism of $R$-modules

$$
 \pi_{ji} : M_j \longrightarrow M_i, 
$$

and if $i \leq j \leq k$, then

$$
 \pi_{ji} \cdot \pi_{kj} = \pi_{ki}. 
$$


\noindent\rule{\textwidth}{0.4pt}


\end{frame}


%---------------------------------------------------
\begin{frame}{2.2 EXAMPLE. }

Let $D(\epsilon)$ be the open disk of centre $0$ and radius $\epsilon$ in $\mathbb{C}$. 

Let $\mathcal{H}(\epsilon) = \mathcal{H}(D(\epsilon))$ be the set of all holomorphic functions defined in $D(\epsilon)$. 

Recall that $\mathcal{H}(\epsilon)$ is a left $A_1$-module. 

Take $\mathbb{R}$ to be the index set. 

If $\epsilon \leq \epsilon'$ in $\mathbb{R}$, then $\mathcal{H}(\epsilon') \subseteq \mathcal{H}(\epsilon)$. 

Hence we may take

$$
 \pi_{\epsilon'\epsilon} : \mathcal{H}(\epsilon') \longrightarrow \mathcal{H}(\epsilon) 
$$

to be the restriction of a holomorphic function in $\mathcal{H}(\epsilon')$ to $D(\epsilon)$. 

This gives us a {\color{red}directed family} of $A_1$-modules.

\noindent\rule{\textwidth}{0.4pt}


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 3
\section{MICROFUNCTIONS.}
%---------------------------------------------------
\begin{frame}{3.0. DEFINITION. }

Let $D(\epsilon)$ be the open disk of $\mathbb{C}$ of centre $0$ and radius $\epsilon$. 

Let $D{\,}'(\epsilon) = D(\epsilon) \setminus 0$.

The {\color{red}{universal cover}} of $D{\,}'(\epsilon)$ is the set 
$\tilde{D}(\epsilon) = \{z \in \mathbb{C} : \text{Re}\, (z) < \log(\epsilon)\}.$ 

The projection $\pi$ of $\tilde{D}(\epsilon)$ on $D{\,}'(\epsilon)$ is defined by $\pi(z) = \exp(z)$. 

Note that $\pi$ is surjective. 
%if $\rho < \epsilon$ is a positive real number, then $\rho e^{i\theta} = \pi(\log(\rho) + i\theta)$, and $\log(\rho) + i\theta \in \tilde{D}(\epsilon)$. 

The relative positions of these sets and maps are shown in the following diagram.

$$
\begin{array}{ccc}
\tilde{D}(\epsilon) & & \\
\pi \downarrow & & \\
D{\,}'(\epsilon) & \longrightarrow & D(\epsilon)
\end{array}
$$

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{3.1 PROPOSITION.}

We make the set $\mathcal{H}(\tilde{D}(\epsilon))$ of functions that are holomorphic in $\tilde{D}(\epsilon)$ into an $A_1(\mathbb{C})$-module. 
Let $h \in \mathcal{H}(\tilde{D}(\epsilon))$. 

The action of a polynomial $f \in \mathbb{C}[x]$ on $h$ is given by $f \cdot h = f(e^x)h(z)$. 

The operator $\partial = d/dx$ acts on $h$ by the formula $\partial \cdot h = h'(z)e^{-z}$. 

{\color{red}
This proposition says that the map

$$
\pi^* : \mathcal{H}(D{\,}'(\epsilon)) \longrightarrow \mathcal{H}(\tilde{D}(\epsilon))
$$

defined by $\pi^*(h)(z) = h(\pi(z))$ is an injective homomorphism of $A_1(\mathbb{C})$-modules.
}

\noindent\rule{\textwidth}{0.4pt}


\end{frame}


%---------------------------------------------------
\begin{frame}{3.2 DEFINITION.}

Since $D{\,}'(\epsilon) \subseteq D(\epsilon)$, we have that $\mathcal{H}(D(\epsilon))$ is a submodule of $\mathcal{H}(D{\,}'(\epsilon))$. 

Since $\pi^* : \mathcal{H}(D{\,}'(\epsilon)) \longrightarrow \mathcal{H}(\tilde{D}(\epsilon))$ is injective, we can consider the quotient module 
$$\mathcal{M}_{\epsilon} := \mathcal{H}(\tilde{D}(\epsilon))/\pi^*(\mathcal{H}(D(\epsilon))).$$ 

If $\epsilon' \leq \epsilon$, then $\tilde{D}(\epsilon') \subseteq \tilde{D}(\epsilon)$. 
Thus $\mathcal{H}(\tilde{D}(\epsilon)) \subseteq \mathcal{H}(\tilde{D}(\epsilon'))$. 

This induces a homomorphism of $A_1(\mathbb{C})$-modules
$$
\tau_{\epsilon'\epsilon} : \mathcal{M}_{\epsilon} \longrightarrow \mathcal{M}_{\epsilon'}.
$$

Hence $\{\mathcal{M}_{\epsilon} : \epsilon \in \mathbb{R}\}$ is a {\color{red}directed family} of $A_1(\mathbb{C})$-modules, and we may take its direct limit. 
This limit, denoted by $\mathcal{M}$, is called the {\color{red}module of microfunctions}.

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{3.3 EXAMPLE.}

The canonical projection of $\mathcal{H}(\tilde{D}(\epsilon))$ onto $\mathcal{M}_{\epsilon}$ is compatible with the limit, and determines a homomorphism of $A_1$-modules
$$
\text{can} : \mathcal{H}(\tilde{D}(\epsilon)) \longrightarrow \mathcal{M}.
$$

The function $e^{-z}/2\pi i$ is holomorphic in $\tilde{D}(\epsilon)$. But $e^{-z}/2\pi i$ is also the image of $1/2\pi iz$ under $\pi^*$. 
However, $1/2\pi iz$ is not holomorphic in $D(\epsilon)$. 

Hence $\text{can}(e^{-z}/2\pi i)$ is a non-zero element of $\mathcal{M}$, it is called the {\color{red}Dirac delta microfunction}, and denoted by $\delta$.

The equation $xh = 0$ has no analytic solution, but it is satisfied by the Dirac delta. 


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\end{frame}

%---------------------------------------------------
\begin{frame}{3.4 EXAMPLE.}

Another important example is the {\color{red}Heaviside microfunction}, defined by $$Y = \text{can}(z/2\pi i). $$

Note that $z/2\pi i$ is the image of $\log(z)/2\pi i$ under $\pi^*$. 

Since $\log(z)/2\pi i$ is not holomorphic in $D(\epsilon)$, the hyperfunction $Y$ is non-zero. 

Moreover, $Y$ is the integral of $\delta$:
$$
\partial \cdot Y = \text{can}(e^{-z}/2\pi i) = \delta.
$$

\noindent\rule{\textwidth}{0.4pt}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 4
\section{EXERCISES. }
%---------------------------------------------------
\begin{frame}{Exercise 4.1. }

Show that the $K$-vector space of polynomial solutions in $K[x_1,x_2]$ of $x_1\partial_2 - x_2\partial_1$ has infinite dimension.

\noindent\rule{\textwidth}{0.4pt}



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%---------------------------------------------------
\begin{frame}{Exercise 4.2. }

Show that the set of polynomial solutions in $\mathbb{C}[x]$ of a differential operator of $A_1(\mathbb{C})$ has finite dimension.


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%---------------------------------------------------
\begin{frame}{Exercise 4.3. }

Let $U$ be an open set of $\mathbb{R}^n$. Using Baire's theorem and the fact that $C{\,}^{\infty}(U)$ is a metrizable vector space show that any basis of $C{\,}^{\infty}(U)$ as a real vector space is uncountable.

\noindent\rule{\textwidth}{0.4pt}

\textbf{Hint:} Suppose the basis $\{v_i : i \in \mathbb{N}\}$ is countable. Let $V_k$ be the subspace generated by $v_1,\ldots,v_k$. Show that $V_k$ is a closed set of $C{\,}^{\infty}(U)$ with empty interior and that $\bigcup_k V_k = C{\,}^{\infty}(U)$. Obtain a contradiction using Baire. A proof that $C{\,}^{\infty}(U)$ is a metrizable vector space is found in [Rudin 91, Ch.1].



\end{frame}

%---------------------------------------------------
\begin{frame}{Exercise 4.4. }

Show that $C{\,}^{\infty}(U)$ is not finitely generated as an $A_n(\mathbb{R})$-module.


\noindent\rule{\textwidth}{0.4pt}

\textbf{Hint:} Suppose that it is generated by $f_1,\ldots,f_k$, and show that $x^{\alpha}\partial^{\beta} \cdot f_i$ forms a countable set of generators for $C{\,}^{\infty}(U)$ as a real vector space, for $\alpha,\beta \in \mathbb{N}^n$ and $1 \leq i \leq k$. This contradicts Exercise 4.3.


\end{frame}

%---------------------------------------------------
\begin{frame}{Exercise 4.5. }

Exercises 4.5 to 4.7 explain the construction of the module of hyperfunctions. This is very similar to the construction of the microfunctions presented in §3.


Let $\Omega$ be an open interval of $\mathbb{R}$. A \textit{complex neighbourhood} of $\Omega$ is an open set $U$ of $\mathbb{C}$ that contains $\Omega$. Consider the set

$$
\mathcal{U} = \{U : U \text{ is a complex neighbourhood of } \Omega\}.
$$

Show that $\mathcal{U}$ is a directed set for the order $\supseteq$.

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{Exercise 4.6. }

Let $\Omega$ be an open interval of $\mathbb{R}$ and $U$ a complex neighbourhood of $\Omega$.
\begin{enumerate}
    \item Show that $\mathcal{H}(U) \subseteq \mathcal{H}(U \setminus \Omega)$ and that both are $A_1(\mathbb{R})$-modules, where a polynomial acts by multiplication and $\partial$ by differentiation.
    \item Show that $\mathcal{H}(U \setminus \Omega)/\mathcal{H}(U)$ is a directed family of $A_1(\mathbb{R})$-modules with respect to the directed set $\mathcal{U}$.
    \item Let $\mathcal{B}(\Omega) = \varinjlim \mathcal{H}(U \setminus \Omega)/\mathcal{H}(U)$ and $h \in \mathcal{H}(U \setminus \Omega)$, for some $U \in \mathcal{U}$. Denote by $[h]$ the image of $h$ in $\mathcal{B}(\Omega)$. Show that if $h$ can be extended to a holomorphic function on $U$ then $[h] = 0$.
    \item Using the notation of the previous item, show that if $f \in \mathbb{R}[x]$ then $f \cdot [h] = [fh]$ and $\partial \cdot [h] = [h']$.
\end{enumerate}

\noindent\rule{\textwidth}{0.4pt}

$\mathcal{B}(\Omega)$ is called the \textit{module of hyperfunctions of} $\Omega$.


\end{frame}

%---------------------------------------------------
\begin{frame}{Exercise 4.7. }

Let $\Omega = (0,1)$. The Heaviside hyperfunction is $Y = [\log(-z)/2\pi i]$ and the Dirac hyperfunction $\delta = [1/2\pi iz]$. Show that $\partial \cdot Y = \delta$.

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{Exercise 4.8. }

Show that the submodule of the module of microfunctions $\mathcal{M}$ generated by $\delta$ is isomorphic to $\mathbb{C}[\partial]$.

\noindent\rule{\textwidth}{0.4pt}


\end{frame}

%---------------------------------------------------
\begin{frame}{Exercise 4.9. }

Let $\delta'$ be the first derivative of the Dirac microfunction. Let $A_1(\mathbb{C})\delta'$ be the submodule of $\mathcal{M}$ generated by $\delta'$. Show that
\begin{enumerate}
    \item $A_1(\mathbb{C})\delta' = A_1(\mathbb{C})\delta$.
    \item $A_1(\mathbb{C})\delta' \cong A_1(\mathbb{C})/J$, where $J$ is the left ideal of $A_1(\mathbb{C})$ generated by $x^2$ and $x\partial + 2$.
\end{enumerate}


\noindent\rule{\textwidth}{0.4pt}

\textbf{Hint:} If $Q \in A_1(\mathbb{C})$ satisfies $Q\delta' = 0$ then we have $Q \cdot \partial \in A_1(\mathbb{C})x$, the annihilator of $\delta$. Write $Q = Q_2x^2 + Q_1x + Q_0$, where $Q_2 \in A_1(\mathbb{C})$, $Q_0, Q_1 \in \mathbb{C}[\partial]$, and calculate $Q \cdot \partial$.


\end{frame}

%---------------------------------------------------
\begin{frame}{Exercise 4.10. }

Let $\delta^m$ be the $m$-th derivative of the Dirac microfunction $\delta$. Show that $A_1(\mathbb{C})\delta^m$ is isomorphic to $A_1(\mathbb{C})/J$, where $J$ is the left ideal of $A_1(\mathbb{C})$ generated by $x^m$ and $x\partial + m$.

\noindent\rule{\textwidth}{0.4pt}

\textbf{Hint:} Induction and Exercise 4.9.

\end{frame}

%---------------------------------------------------
\end{document}

See conflicts as exercises. 

